Cartesian Product is Small

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Theorem

Let $a$ and $b$ be small classes.


Then their Cartesian product $a \times b$ is small:

$\map {\mathscr M} {a \times b}$


Proof

\(\ds a \times b\) \(=\) \(\ds \set {\tuple {x, y}: x \in a \land y \in b}\) Definition of Cartesian Product
\(\ds \) \(=\) \(\ds \set {\tuple {x, y}: \set x \subseteq a \land \set y \subseteq b}\) Singleton of Element is Subset
\(\ds \) \(\subseteq\) \(\ds \set {\tuple {x, y}: \set {x, y} \subseteq a \cup b}\) Set Union Preserves Subsets
\(\ds \) \(\subseteq\) \(\ds \set {\tuple {x, y}: \tuple {x, y} \subseteq \powerset {a \cup b} }\) Power Set of Subset and Subset Relation is Transitive



So by definition of power set:

$a \times b \subseteq \powerset {\powerset {a \cup b} }$


By Union of Small Classes is Small, $a \cup b$ is small.


By the Axiom of Powers, $\powerset {\powerset {a \cup b} }$ is small.


By Axiom of Subsets Equivalents, $a \times b$ is small.

$\blacksquare$


Sources